Taper, volume, and bark thickness models for spruce, pine, and birch in Norway

ABSTRACT Taper models, which describe the shape of tree stems, are central to estimating stem volume. Literature provides both taper- and volume models for the three main species in Norway, Norway spruce, Scots pine, and birch. These models, however, were mainly developed using approaches established over 50 years ago, and without consistency between taper and volume. We tested eleven equations for taper and six equations for bark thickness. The models were fitted and evaluated using a large dataset covering all forested regions in Norway. The selected models were converted into volume functions using numerical integration, providing both with- and without-bark volumes and compared to the volume functions in operational use. Taper models resulted in root mean squared error (RMSE) of 7.2, 7.9, and 9.0 mm for spruce, pine, and birch respectively. Bark thickness models resulted in RMSE of 2.5, 6.1, and 4.1 mm, for spruce, pine, and birch respectively. Validation of volume models with bark resulted in RMSE of 12.7%, 13.0%, and 19.7% for spruce, pine, and birch respectively. Additional variables, tree age, site index, elevation, and live crown proportion, were tested without resulting in any strong increase in predictive power.


Introduction
Tree shape and stem taper are affected by genetics, growing conditions, and competition among trees (Saarinen et al. 2020).The ability to mathematically describe the shape of tree stems has been a continuous work for more than a hundred years (Kublin et al. 2013;McTague and Weiskittel 2021).In addition to providing estimates of stem diameters at any point along the stem (West 2015), taper functions allow the estimation of stem volume, which is a key variable for forest inventories, management, and planning (McTague and Weiskittel 2021).Taper models also enable the estimation of log volumes and minimum top diameters for logs in different quality classes.In combination with growth and yield models, taper models can be used to estimate timber assortment yields at different growing sites and alternative management schedules (de-Miguel et al. 2012).
The shape of tree stems can be described as paraboloid, with a neiloid form at the bottom and a cone form at the top.Different equations and statistical techniques have been used to describe this shape, from simple polynomials to nonlinear and multivariate regression, and segmented equations.Most of the work on taper modeling has traditionally used parametric approaches, but there are also examples of semi-or nonparametric techniques (e.g.Robinson et al. 2011;Scolforo et al. 2018).Generally, variable-form parametric equations have been found to be advantageous (Rojo et al. 2005;Gómez-García et al. 2013;McTague and Weiskittel 2021).
Variable-form equations have mainly been developed as fixed-effects nonlinear equations.However, the data used to develop taper models are multiple measurements on individual stems, and the measurements are therefore not fully independent.To account for tree-specific error correlation structures, mixed-effects modeling is often used (Muhairwe et al. 1994;Gregoire and Schabenberger 1996;Fang and Bailey 2001;Schröder et al. 2014).Mixed-effects models can also be used to analyze the effects of additional stand and site variables that can affect the stem form (Muhairwe et al. 1994;Fang and Bailey 2001;Pukkala et al. 2019).However, the application of mixed-effects models requires additional diameter measurements, which are usually not part of common forestry practice.Application of a mixed-effects model without the inclusion of random effects has been found to result in less precise predictions compared to ordinary least squares approaches (Arias-Rodil et al. 2015).
The groundwork for the taper-and volume models currently used in Norway was laid down a century ago by research organized by the Norwegian Forest Research Institute, and works on the taper-and volume of Norway spruce and Scots pine were published in a series of communications in the 1920s (Archer 1920;Eide 1922Eide , 1923aEide , 1923bEide , 1925Eide , 1927;;Langsaeter 1927).The models for spruce, presented as tables, for diameter and stem volume without bark at different heights above the stump were published in 1929 (Eide and Langsaeter 1929) and later updated with additional data (Eide and Langsaeter 1954).The models required the input of diameter at breast height without bark and tree height above the stump and were fitted using Behre's formula (A1).The fundamental structure of the applied stem taper models in Norway today is based on the work of Strand (1967).Strand developed taper models for pine, consisting of a set of nine separate polynomials (A2), each used to predict the diameter at a relative height of the stem.In order to predict the diameter at different heights, interpolation must be applied.Gjølberg (1978) developed taper models for spruce based on the tables presented by Eide andLangsaeter (1929, 1954) and following the procedure in Strand (1967).For birch, corresponding models were developed by Blingsmo (1985), who also followed the same procedure as Strand (1967).Although taper models can be integrated to provide volume, separate single tree volume models were developed for Norway in the 1960s (Braastad 1966;Brantseg 1967;Vestjordet 1967).The models were developed as polynomial regression models for both with-and without bark volumes (A3-A15).These models are currently still in use and provide estimates from single trees to the national level (e.g.Breidenbach et al. (2020); Rahlf et al. (2021)).For comparison and analysis of the models developed in the present study, this collection of models is termed the conventional models.
Acknowledging the shortcomings of earlier work and having several separate models for taper and volume, Pukkala et al. (2019) developed new taper models for spruce in Norway using a small part of the same data (477 trees) as Eide and Langsaeter (1954).Pukkala et al. (2019) evaluated models based on equations that had shown good performance in previous studies, B00 (Bi 2000), K95 (Kozak 1997 (equation 3)), and K02 (Kozak 2004 (equation 4)).Further, Pukkala et al. (2019) also investigated the effects of additional information on site index and age, finding these to yield taper models with significantly less error.Although these models were found to provide significant improvements to the existing models, they were developed from data from a limited geography and were only available for spruce.
Taper models can be developed with-and without bark, provided that data on bark thickness along the stem is available.Models without bark can be developed separately from models with bark, but because the relationship is correlated, modeling approaches could benefit from the relationship and take this into account (Gregoire and Schabenberger 1996).Developing the models simultaneously should also ensure consistent results for with-and without-bark predictions.Another option to ensure consistent predictions is to develop separate bark models which can be used to subtract the bark from predictions from models with bark.In Norway, the taper models by Gjølberg (1978), Strand (1967), andBlingsmo (1985) can be used together with models for double bark developed by Vestjordet (1967), Brantseg (1967), andBraastad (1966) for spruce, pine, and birch, respectively.
The objective of the present study is to develop taper models, and subsequent volume models, with-and without bark, for the three main species in Norway, Norway spruce (Picea abies (L.) Karst), Scots pine (Pinus sylvestris L.), and birch (Betula pubescens Ehrh.and B. pendula Roth), based on a large dataset representing the full gradient of climatic growing conditions in Norway.Taper modeling was done by testing eleven equations on a large national dataset (3549 spruce trees, 11,382 pine trees, and 3901 birch trees).Taper models were developed as nonlinear least squares regression models.To account for site and competition effects in the models, variables related to site index, age, live crown proportion, and elevation were evaluated.To model stem taper without bark, the development of separate models for double bark thickness followed the same procedure as the taper modeling.Six bark thickness model forms were evaluated on 671 spruce trees, 3642 pine trees, and 1137 birch trees.The resulting taper models were integrated into volume models compared to observed volumes and existing taper-and volume models withand without bark on a subsample of the dataset.

Data
The stem data used in the present study were collected as part of NIBIO's (Norwegian Institute of Bioeconomy Research) longterm research trials established in the early twentieth century to study the effects of silvicultural treatments.The trial plots are distributed throughout Norway, and all plots used in the present study were established to study the effects of thinning on volume production, mortality, stability, damages, and climatic effects (Andreassen et al. 2018).During remeasurements, the diameter at breast height, 1.3 m above ground of all trees is measured.For additional measurements, trees are systematically sampled, and a number of variables, e.g. total tree height, tree height above the stump, and crown height, are registered.Diameters for taper assessment are measured from a selection of felled trees.Diameters and bark thickness were, for the main part of the dataset, measured crosswise at 0.5 and 1.3 m and along the stem in one-meter or two-meter intervals to the top, starting at 1.5 and 2 m, respectively.Additionally, a number of variables were registered at the tree level.Total tree age was assessed by counting tree rings at the stump and estimating the age of the stump (Braastad 1980).Site index, describing the productivity of the site, is calculated based on dominant tree height at 40 years of age (Tveite 1977;Tveite and Braastad 1981).Crown height, determined by the height of the lowest green whirl, was used to calculate a relative live crown proportion by dividing it by total tree height.The additional effects were only available for a subset of the data (Table 3).
For the present study, a total of 18,832 trees from 321 field plots were used.The trees were destructively sampled on remeasurement occasions from 1915 until 1992.The majority of the data has never been converted into a digital format and was only available in written field books.Based on all the available data, a selection of trials with taper measurements was selected to cover all three main tree species and have a strong geographic coverage of the country (Figure 1).Trees on field plots in the region of Western Norway had been digitized for earlier studies, but diameters were only digitized at 1.3 m above the ground and from 1.5 to 6.5 m in onemeter intervals (Figure 2, Table 1).Further, no bark measurements were available in the data from Western Norway   (Table 2).In the other regions, the frequency of trees with bark measurements varied among regions and tree species and had an average of 38% for spruce, 44% for pine, and 29% for birch (Table 2).On the tree level, the frequency was largely determined by the total tree height (Figure 3).The dataset was thoroughly cleaned for registration and digitization errors.283 trees were removed from the dataset after visual inspection of plotted data and deviances from a curve acquired by smoothing.For the bark modeling, two additional spruce trees and two pine trees were removed.Further, all registrations of double bark larger than the diameter with bark were discarded.The measurements of height and diameter were for about half of the tree taken above the stump instead of above the ground.For these trees, the height of the measurements was converted by adding one percent of the total tree height to the height measurements.Thus, tree height was defined as the total tree height above the ground.

Taper models
Eleven taper equations were selected from prior studies based on their performance.We also sought equations that were developed for the same, or similar, species as in the present study.These equations are referred to as L82 (A16) (Laasasenaho 1982), K88 (A17) (Kozak 1988), K95 (A18) (Kozak 1997), M99a (A19) and M99b (A20) (Muhairwe 1999), Z99 (A21) (Zakrzewski 1999), B00 (A22) (Bi 2000), K02 (A23) (Kozak 2004), SZ04 (A24) (Sharma and Zhang 2004), G09 (A25) (Goodwin 2009), and K13 (Kublin et al. 2013).The fitted models were evaluated in a five-fold cross-validation procedure.Models were evaluated using root mean squared error (RMSE) and a mean absolute difference (MAD).The RMSE was calculated as the square root of the mean of the square of the difference between the observed and predicted values (1).The MAD was calculated as the mean of the absolute difference between the mean of the predicted values and the mean of the observed values (2).Ten of the eleven equations were fitted as nonlinear least squares models using a Levenberg-Marquardt algorithm implemented in the minpack.lmpackage (Elzhov et al. 2016) in R (R Core Team 2020).The K13 model is a result of a different modeling approach using cubic spline regression and incorporating mixed effects on tree level (Kublin et al. 2013).The model was fitted using the package TapeR (Kublin and Breidenbach 2022)  where n is number of observations, y i is the observed value for the ith observation, y i is the predicted value for the ith observation.

Bark thickness models
Models for double bark thickness were developed in correspondence with the approach used for taper modeling.Stängle et al. (2017) evaluated six equations used to estimate double bark thickness, and these were evaluated in the present study to develop models for double bark thickness for spruce, pine, and birch.The Z74 equation (A26) was introduced by Loetsch et al. (1973) and applied by Zacco (1974) in the standards for forest machine data and communication (Arlinger et al. 2012;SKOGFORSK 2020).The H04 (A31) (Hannrup 2004) equation is also used in the same standards in Sweden.Wilhelmsson et al. (2002) developed models for both pine and spruce in Sweden based on a similar equation (W02 [A30]).The CP86 equation (A29) was originally developed for tropical pines (Cao and Pepper 1986), but performed on spruce in Germany as well (Stängle et al. 2017).The G83a and G83b equations (A27 and A28) were developed on pines in New Zealand (Gordon 1983) but performed well in the study by Stängle et al. (2017), where the G83b equation resulted in the smallest errors.The W02 and G83b equations originally applied a log-transformed response.In the present study, we applied a nonlinear least squares regression technique, and the response was therefore not transformed.Instead, an exponential transformation was applied to the right-hand side of the equations.

Including site and environmental factors as predictors
The additional site and environmental factors, site index, age, live crown proportion, and elevation, were continuous variables.For each of these factors, models were fitted using the selected taper equation K88 (A17) to evaluate the effect of including the factors in the modeling of stem taper.Following the approach by Pukkala et al. (2019), we attempted to make all parameters dependent on the additional variable by multiplying each parameter with a second parameter, incorporating additional information.For age, e.g.b 1 in K88 was replaced by (b 01 + b 02 • Age).However, for several of the factors, this approach resulted in a lack of convergence when fitting the models.Instead of adding the effect to all predictor variables, we placed the additional parameter only on the b 1 parameter.Models with additional effects were compared to models without additional effects by analysis of variance and AIC values, and RMSE and MAD values from five-fold cross-validations.
The effect of the additional factors was expressed as a relative change in AIC, RMSE, and MAD, denoted ΔAIC, ΔRMSE, and ΔMAD, respectively.This change was calculated by subtracting the respective values of models without additional effects from the values of models with additional effects, divided by the values of the models without additional effects.

Comparisons of observed volumes, and predictions from current and conventional models
The resulting taper models were transformed into volume models with bark using numerical integration in R. By subtracting the modeled bark thickness from the taper model and integrating the result, a volume model without bark was found.The volume models were compared to a calculated observed volume.A subset of trees with complete registrations for bark and height above stump and at least one diameter measurement per meter was used for comparing observed volume and model predictions from current and conventional models.The observed volume was calculated and summarized at tree level for each log section using Smalian's formula (3).The observed volume was limited to the lowest diameter measurement of 0.5 m above ground, and compared to model predictions with volume predicted from 0.5 m above ground.
where V obs is total volume of a tree where d i is diameter at height h i for diameter measurement i, n is the number of measurements, h top is total tree height.Measurements are sorted in ascending order with respect to h.The conventional models by Vestjordet (1967) (A11-A12), Brantseg (1967) (A6-A7), and Braastad (1966) (A3), predict volumes above stump height (1% of tree height) and could therefore not be compared to observed volumes.Instead, to evaluate them consistently, the conventional model predictions were compared to volumes derived from numerically integrating the new taper models between stump and total tree height, on the same dataset.The conventional model predictions without bark were made by using the double bark functions of Vestjordet (1967), Brantseg (1967), andBraastad (1966), to subtract bark from the observed diameter at breast height with bark.

Taper models
For selecting a preferred taper equation, eleven candidate equations were fitted as nonlinear least squares equations and evaluated by the RMSE and MAD resulting from fivefold cross-validation.All equations performed well, with RMSE value ranges of 0.72-1.06cm for spruce, 0.78-1.12cm for pine, and 0.70-1.06cm for birch (Table 4).Correspondingly, MAD values ranged from 0.01-0.18cm for spruce, 0.00-0.15cm for pine, and 0.00-0.14cm for birch.In line with previous studies, the accuracy is generally better with a larger number of parameters (Rojo et al. 2005;Schröder et al. 2014).The K88 and K02 equations are frequently used and performed well in the present study.When comparing models, the K88 model had the smallest RMSE value for spruce and the third smallest for pine and birch.It also had small MAD values for all tree species, with the second smallest for spruce and pine and the third smallest for birch.The K02 model performed well with the second smallest RMSE value for all three species.The MAD values for K02 were larger compared to the K88 model for all tree species.Based on these results and that the K88 model is a simpler model with only eight parameters compared to the K02 with nine, we selected the K88 model for further analysis.

Bark models
Similarly to the taper modeling, six equations were selected and fitted for double bark thickness.Cross-validation of the resulting models produced RMSE values at 2.5 mm for spruce and ranged from 3.9 to 6.5 mm for pine, and from 3.3-4.1 mm for birch.MAD values ranged from 0.0-0.02mm for spruce, 0.0-0.2mm for pine, and 0.0-0.1 mm for birch.As with the taper models, the models with more parameters and predictor variables generally resulted in increased accuracy, although not for spruce.The G83a model performed well for spruce and birch with only diameter as the input variable (Table 5).The G83b model with diameter, height, and total tree height as predictor variables resulted in the smallest RMSEs for all three species.The MAD values, however, were larger for the G83b model compared to most of the other equations evaluated.Furthermore, three of the parameters were not significant, and the parameter estimates indicated multicollinearity between the variables.Therefore, the H04 equation (A31) was selected for further analysis as it had an overall good performance with low MAD values and only three parameters.
Fitted models for double bark thickness (B) in mm for spruce (7), pine (8), and birch ( 9), where D is diameter at breast height in cm and d is the diameter in cm.

Analysis of additional effects on taper
To evaluate the effect of additional site variables on tree taper, separate models were fitted with site index, age, live crown proportion, and elevation, and compared to models fitted without these effects.The results show that adding information on site index, age, live crown proportion, and elevation in most cases improves the model performance.
For spruce, site index, age, and elevation resulted in models with significantly smaller errors and reduced AIC values.For pine, all additional variables were found to be significant, and for birch site index, live crown proportion, and elevation were significant.The AIC values were also reduced in all cases except for birch age.The cross-validation revealed that the results were less clear as some of the additional effects resulted in increased MAD values.This was the case for live (4) crown percentage for all tree species, elevation for spruce and pine, and site index and age for birch.For site index, which showed the largest reduction, percent-wise, in AIC and RMSE for spruce and pine, the MAD values were also reduced (Table 6).For spruce and pine, the inclusion of site index resulted in a reduced RMSE of 0.56 and 3.22 percent of the mean, respectively.Although it also resulted in reduced accuracy, this increase was from very small values.
For birch, the largest reduction in RMSE was found by including live crown proportion.

Comparisons of observed volumes and predictions from current and conventional models
Comparisons of observed volumes and current and conventional model predictions were performed.The comparisons were made on a subset of the data that had complete registrations of stump heights and bark thickness and more than one diameter measurement per meter of total tree height.
In total this summed to 3850 trees (483 spruce, 2634 pine, 733 birch).The comparison of observed volumes, with and without bark, and current model predictions was performed down to 0.5 m tree height and resulted in relative RMSE (RMSE divided by the mean of the observed values) for volumes with bark of 12.7%, 13.0%, and 19.7% for spruce, pine, and birch, respectively.Results without bark showed similar accuracy with relative RMSEs of 12.9%, 13.5%, and 19.7% for spruce, pine, and birch respectively.Plots of observed vs. predicted volumes, residuals in cubic meters (observed minus predicted volumes), and residuals in percentage of the observed volume, all with bark, show that there is a high correlation between observed and predicted volumes both with-and without bark (Figures A1 and A2).Further, the residual plots in Figures A1 and A2 show that the biases of the models are small for both with-and without-bark volumes.
Comparisons between volume predictions from the current and conventional models are shown in Figures A3  and A4.These plots clearly show that there are differences in the models, but also that the predictions for spruce and pine are relatively similar.The residual plots further illustrate that the conventional models for spruce and pine are constructed by two separate functions for trees below and above 12 cm in diameter at breast height.This can be seen most clearly for pine but can be observed for spruce as well (Figure A3).
To further illustrate the difference between the models, plots showing the conventional model minus the current model were produced (Figure 4).The stump height was set to 1% of the tree height.The overlaid point scatter of the trees used in the present study shows that, for trees with diameter at breast height to height relationship in the dataset, the current models generally predict less volume for all tree species.However, the plots also show that the differences are dependent on the diameter at breast height to tree height relationship.
Observed bark volume as a percentage of total volume was calculated on the subset of the data that had complete registrations and compared to predicted bark percentages from the current and conventional models (Table 7).However, the results showed that the conventional models predicted bark volume closer to the observed bark volume for both spruce and pine.

Discussion
A lot of effort has been invested into describing the shape, and consequently also the volume, of tree stems over the past century.Taper models are now the primary means to estimate stem volume (McTague and Weiskittel 2021).The models for stem taper, volume, and bark thickness in Norway were developed over the last century in separate studies, not always considering the consistency between taper, volume, and bark thickness.The models also had other shortcomings related to the development of modern statistics and computational power.The development of more flexible and coherent taper-and volume models was therefore expected to yield less errors.In the present study, several model forms were selected and evaluated, for both stem taper and bark thickness, on a large dataset covering forested areas in Norway.
In line with previous research on stem taper, e.g.Rojo et al. (2005), results from the procedures of selecting a taper equation support the choice of a variable-form equation.All the evaluated equations performed satisfactorily and in line with the trend of increased accuracy with an increased number of parameters.The K88 model, with eight parameters respectively, was selected for further analysis based on the overall performance for all species in the present study.However, as stated by de-Miguel et al. (2012), the relationship between the number of parameters and model error is not straightforward.Most noteworthy is that the relatively simple equation of Zakrzewski (1999), with only two parameters, performed quite well.Further, the equation of Goodwin (2009) performed especially well for birch.This equation also has the quality of being flexible to diameter input, meaning that other diameters than diameter at breast height can be used.This property is only shared with the K13 equation (Kublin et al. 2013) among the equations evaluated in the present study.
The results for bark equations were in line with Stängle et al. (2017) who found that the flexible equation proposed by Gordon (1983) resulted in the largest accuracies in a study on spruce bark thickness in Germany.In the present study, the equation (G83b) also resulted in the largest accuracy for all three species.However, the results from cross-validation showed the model to be less precise than models with fewer parameters and predictor variables (Table 4).The proportion of bark to total volume varies with the size of the trees, where smaller trees generally have a larger proportion of bark volume to total volume.In the present study, bark percentage was calculated for a subset of trees and compared to Spruce 483 16.9 (1.9-51.2) 14.6 (7.7-48.5)14.1 (10.9-39.0)14.5 ( both current and conventional models.Bark volume seems to vary a lot depending on tree size and sites.In Finland, Laasasenaho et al. ( 2005) found a bark percentage of 17 for spruce with a mean diameter at breast height of 18 cm.Whilst, the study of Stängle et al. (2017) found percentages of 10.8 and 9.8 on two datasets for spruce in Germany.The tree size in the data was larger relative to the present study with mean diameter at breast height of 34.5 and 39.8 cm respectively, explaining the smaller volume percentages compared to the present study.
The data consist of trees from experimental plots established for scientific research in productive forests (Braastad 1980).All plots in the present study were originally established to study the effects of thinning and in forest stands established after a clear-cut (Andreassen et al. 2018).Therefore the models, developed from these data, will likely be most accurate and precise in forests with similarly silvicultural treatments.The majority of the plots were established as monoculture stands with one dominant species.In natural or semi-natural forests, with different stand structures, species compositions, and silvicultural management schemes, e.g.selective cutting, the models will likely be less accurate and precise.
Inclusion of additional effects, site index, age, live crown proportion, and elevation, generally results in models with significantly smaller errors.However, the effects were in most cases relatively small.The largest effect on RMSEs from cross-validated results was found for site index for spruce and pine, and for live crown proportion for birch.Pukkala et al. (2019) evaluated the effects of age and site index and found that although including information on age and site index resulted in significant improvements to a spruce model, the effect was small.Muhairwe et al. (1994) evaluated the effects of site, live crown proportion, and age on the K88 function and concluded that, with the exception of live crown proportion, the additional effects resulted in minor improvements.Results from the present study indicate that including site index in the taper models results in increased accuracy.One drawback is that the availability of registrations limits the number of observations.

Conclusions
Models of stem taper and volume are important tools in forest management enabling estimation of log volumes, quality classification, and growth and yield forecasting.The models in current use in Norway are a combination of several models developed over the last century.In the present study models for estimating taper-and volume of tree stems, with-and without bark, were developed for Norway spruce, Scots pine, and birch using a large national dataset.From a total of eleven equations for stem taper and six equations for bark thickness, the equations suggested by Kozak (1988) and Hannrup (2004) were selected for stem taper and bark thickness respectively.The fitted models can be used for predicting diameters at any point along the stem, stem-and log volume, with-and without bark.The models are presented in the r-package taperNOR (Rahlf and Hansen 2022).

Figure 1 .
Figure 1.Map showing the number of trees in municipalities, and regions, used in the present study.

Figure 2 .
Figure 2. Number of diameter measurements per tree.
Notes: ns = not significant, * = p < 0.05, ** = p < 0.01, *** = p < 0.001, N = number of observations, ΔAIC = change in AIC (Akaikes information criterion) in percentage of AIC without added effect, ΔRMSE = change in RMSE (root mean squared error) in percentage of RMSE without added effect, ΔMAD = change in MAD (mean absolute difference) in percentage of MAD without added effect.

Figure 4 .
Figure 4. Difference between predicted volumes with bark of conventional minus current models in percentage of old model volume.Number of observations from the training data set is overlaid.

Figure A2 .
Figure A2.Comparisons of observed tree volume without bark and current model without bark.

Figure A3 .
Figure A3.Comparisons of conventional vs. current predictions with bark.

Figure A4 .
Figure A4.Comparisons of conventional vs. current model predictions without bark.

Table 1 .
Number of trees and plots in geographical regions used in the present study.

Table 2 .
Number of trees and plots on geographical regions with bark measurements used in the present study.

Table 3 .
Descriptive statistics of tree measurements.Number of trees (n), mean and standard deviation of diameter at breast height (D, sD), mean and standard deviation of tree height (H, sH), mean and standard deviation of site index (SI, sSI), mean and standard deviation of age (Age, sAge), mean and standard deviation of live crown proportion (LCP, sLCP), and mean and standard deviation of elevation (Elev, sElev).
Figure 3. Number of bark measurements per tree for a subset of data with bark measurements.

Table 4 .
Fitting statistics (root mean squared error (RMSE) and mean absolute difference (MAD)) for taper models validated in a five-fold cross-validation.

Table 6 .
Results of taper models fitted using a subset of the data where the additional variables (site index, age, live crown proportion, elevation) were available.

Table 5 .
Fitting statistics for models of double bark thickness (B) in mm, validated in a five-fold cross-validation.Root mean square error (RMSE) and mean absolute difference (MAD) in mm.